where \(C_Q = 1+ O_N(Q^{-N})\) , and \(Q = B^{1/2}\) . Here \(I_{q,\mathbf {a}}(\mathbf {c})\)  and \(S_{q,\mathbf {a}}(\mathbf {c})\)  are exponential sums and integrals defined analogously to [1]}, via
\(\begin{split}S_{q,\mathbf {a}}(\mathbf {c}) &= \sum _{\begin{array}{c}k \bmod q\\ \gcd (k,q)=1\end{array}} \sum _{\mathbf {b} \bmod q}e_q(kF_{\mathbf {a}}(\mathbf {b}) + \mathbf {b}\cdot \mathbf {c}),\\I_{q,\mathbf {a}}(\mathbf {c}) &= \int _{\mathbb {R}^4}w(P_1^{-1}x_1, \ldots , P_4^{-1}x_4)h\left(\frac{q}{Q}, \frac{F_{\mathbf {a}}(\mathbf {x})}{Q^2}\right)e_q(-\mathbf {c}\cdot \mathbf {x})\,\mathrm {d}\mathbf {x}.\end{split}\) 
